Question

two tangents are drawn from the point -2,1 to the parabola y2 =4x if theta is angle between tangent then tan theta
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Answer 1

Answer:

tan θ = 3

Step-by-step explanation:

Given:

• Two tangents are drawn from the point (-2,1)
• Equation of the parabola = y² = 4x
• θ is the angle between the tangents

To Find:

• tan θ

Solution:

Here equation of the parabola is of the form y² = 4ax = y² = ax

Comparing we get the value of a as 1.

Now equation of tangent to a parabola is given by,

Equation of tangent = y = mx + a/m

where m is the slope of the tangent

Here a = 1, hence,

Equation of tangent = y = mx + 1/m

Given that the tangents are drawn from the point (-2, 1) ie, this point lies on the tangent.

Therefore,

1 = -2m + 1/m

Cross multiplying,

1m = -2m² + 1

2m² + m - 1 = 0

Factorising by splitting the middle term,

2m² -m + 2m - 1 = 0

m (2m - 1) + 1 (2m - 1) = 0

(2m - 1) (m + 1) = 0

Either

m + 1 = 0

m = -1

Or

2m - 1 = 0

m = 1/2

Hence the slopes of the tangent are -1 and 1/2

Let us take them as m₁ and m₂

Now tan θ between two lines is given by,

$\tt tan\: \theta=\bigg | \dfrac{m_2-m_1}{1+m_1m_2}\bigg|$

Substitute the data,

$\tt tan\: \theta=\dfrac{\dfrac{1}{2} +1}{1+\dfrac{1}{2}\times -1}$

$\tt tan\: \theta=\dfrac{3}{2}\div \dfrac{1}{2}$

$\tt tan\: \theta=3$

Hence the value of tan θ is 3.

Answer 2

$\Large\bf{\color{cyan}GiVeN,}$

• Two tangents are drawn from the point (-2 , 1) to the parabola $\bf{y^2\:=\:4x}$.

☆ Equation of parabola,

$\orange\bigstar\:\:\bf\blue{y^2\:=\:4ax}$

☆ According to the question,

• a = 1

☆ Equation of tangent,

$\green\bigstar\:\:\bf\purple{y\:=\:mx\:+\:\dfrac{a}{m}\:}$

$\longmapsto\:\:\bf{y\:=\:mx\:+\:\dfrac{1}{m}\:}$

$\bf\red{Here,}$

• x = -2

• y = 1

$\longmapsto\:\:\bf{1\:=\:-2m\:+\:\dfrac{1}{m}\:}$

$\longmapsto\:\:\bf{m\:=\:-2m^2\:+\:1\:}$

$\longmapsto\:\:\bf{2m^2\:+\:m\:-\:1\:=\:0\:}$

$\longmapsto\:\:\bf{2m^2\:+\:2m\:-\:m\:-\:1\:=\:0\:}$

$\longmapsto\:\:\bf{2m\:(m\:+\:1)\:-\:1\:(m\:+\:1)\:=\:0\:}$

$\longmapsto\:\:\bf{(2m\:-\:1)\:(m\:+\:1)\:=\:0\:}$

$\longmapsto\:\:\bf{m\:=\:\dfrac{1}{2}\:~~~or~~~\:m\:=\:-1\:}$

$\bf\pink{Let,}$

• $\bf{m_1\:=\:\dfrac{1}{2}}$

• $\bf{m_2\:=\:-1}$

$\purple\bigstar\:\:\bf{\color{peru}\tan{\theta}\:=\:\mid\:{\dfrac{m_1\:-\:m_2}{1\:+\:m_1\:m_2}}\:\mid}$

$:\implies\:\:\bf{\tan{\theta}\:=\:\mid\:{\dfrac{0.5\:-\:(-1)}{1\:+\:0.5\:(-1)}}\:\mid}$

$:\implies\:\:\bf{\tan{\theta}\:=\:\mid\:{\dfrac{1.5}{1\:-\:0.5}}\:\mid}$

$:\implies\:\:\bf{\tan{\theta}\:=\:\mid\:{\dfrac{1.5}{0.5}}\:\mid}$

$:\implies\:\:\bf\orange{\tan{\theta}\:=\:3\:}$